The variational methods in the likelihood-free setting are generally not designed for posteriors with multimodal structures. Thus they may not render a good approximation of complex posteriors. Considering this, we introduce an adaptation of the mixture population Monte Carlo (MPMC) method proposed by [1]} to the likelihood-free setting. MPMC finds the optimal \(q(\theta )\)  by minimizing \(\mathrm {KL}(\pi \Vert q)\) 
within a class of mixture distributions, such as Gaussian mixtures. The proposed MPMC algorithm in this paper is able to fit posteriors with multimodal and heavy-tailed structures by using a proper mixture distribution. Additionally, MPMC updates iteratively the parameters of mixture distributions by IS computations rather than SGD algorithms. In other words, MPMC does not have to calculate the gradient of KL, which is difficult in the likelihood-free setting. In the original form of the MPMC method [1]}, the number of mixture components \(D\)  is fixed throughout. For practical problems, how to choose a proper \(D\)  in advance is still not clear.
Many mixture parameters need to be initialized when \(D\)  is large.
To handle these issues, we propose an adaptive procedure to update \(D\)  automatically. Meanwhile, we provide a strategy to initialize a new mixture component when updating the number of mixture components. The whole process of our proposed method is automatic with minimal inputs.
The main contributions of this paper are:
